icon

**2-category theory**
## Definitions
* 2-category
* strict 2-category
* bicategory
* enriched bicategory
## Transfors between 2-categories
* 2-functor
* pseudofunctor
* lax functor
* equivalence of 2-categories
* 2-natural transformation
* lax natural transformation
* icon
* modification
* Yoneda lemma for bicategories
## Morphisms in 2-categories
* fully faithful morphism
* faithful morphism
* conservative morphism
* pseudomonic morphism
* discrete morphism
* eso morphism
## Structures in 2-categories
* adjunction
* mate
* monad
* cartesian object
* fibration in a 2-category
* codiscrete cofibration
## Limits in 2-categories
* 2-limit
* 2-pullback
* comma object
* inserter
* inverter
* equifier
## Structures on 2-categories
* 2-monad
* lax-idempotent 2-monad
* pseudomonad
* pseudoalgebra for a 2-monad
* monoidal 2-category
* cartesian bicategory
* Gray tensor product
* proarrow equipment

Let $C,D$ be 2-categories and $F,G:C\to D$ be functors. An **icon** $\alpha:F\to G$ consists of the following:

- the assertion that $F$ and $G$ agree on objects.
- for each morphism $u:x\to y$ in $C$, a 2-cell $\alpha_u:F(u) \to G(u)$ in $D$ (note that this only makes sense because $F$ and $G$ agree on objects, so that $F(u)$ and $G(u)$ are parallel.
- for each 2-cell $\mu:u\to v$ in $C$, we have $\alpha_v . F(\mu) = G(\mu).\alpha_u$.
- for each object $x$ of $C$, $\alpha_{1_x}$ is an identity (modulo the unit constraints of $F$ and $G$, if they are not strict functors).
- for each composable pair $x\overset{u}{\to} y \overset{v}{\to} z$ in $C$, we have $\alpha_v {*} \alpha_u = \alpha_{v u}$ (modulo the composition constraints of $F$ and $G$, if they are not strict functors).

If $D$ is a strict 2-category (or at least strictly unital), then an icon is identical to an oplax natural transformation whose 1-cell components are identities. In general, there is a bijection between icons and such oplax natural transformations, obtained by pre- and post-composing with the unit constraints of $D$. The name “icon” derives from this correspondence: it is an Identity Component Oplax Natural-transformation.

Icons have technical importance in the theory of 2-categories. For instance, there is no 2-category (or even 3-category) of 2-categories, functors, and lax or oplax transformations (even with modifications), but there is a 2-category of 2-categories, functors, and icons. (In fact, this 2-category is the 2-category of algebras for a certain 2-monad.)

Additionally, if monoidal categories are regarded as one-object 2-categories, then monoidal functors can be identified with 2-functors, and monoidal transformations can be identified with icons.

- Stephen Lack,
*Icons*. arxiv:0711.4657

Revised on September 3, 2010 20:45:10
by Toby Bartels
(173.190.153.41)